Fixed point sets through iteration schemes

We introduce the notion of a general fixed point iteration scheme to unify various fixed point iterations in the literatures, and extend the concept of virtual stability of a selfmap to an iteration scheme to obtain a connection, through an explicit retraction, between the convergence set of the scheme and its fixed point set. Moreover, we illustrate how to apply our results to obtain a new criterion for contractibility of the fixed point set of a certain quasi-nonexpansive selfmap.